Optimal. Leaf size=107 \[ \frac {F_1\left (\frac {1-m}{2};1,-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \cot (e+f x))^m \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)} \]
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Rubi [A]
time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3755, 3751,
525, 524} \begin {gather*} \frac {\tan (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \left (\frac {b \tan ^2(e+f x)}{a}+1\right )^{-p} F_1\left (\frac {1-m}{2};1,-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right )}{f (1-m)} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 3751
Rule 3755
Rubi steps
\begin {align*} \int (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p \, dx &=\left ((d \cot (e+f x))^m \left (\frac {\tan (e+f x)}{d}\right )^m\right ) \int \left (\frac {\tan (e+f x)}{d}\right )^{-m} \left (a+b \tan ^2(e+f x)\right )^p \, dx\\ &=\frac {\left ((d \cot (e+f x))^m \left (\frac {\tan (e+f x)}{d}\right )^m\right ) \text {Subst}\left (\int \frac {\left (\frac {x}{d}\right )^{-m} \left (a+b x^2\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left ((d \cot (e+f x))^m \left (\frac {\tan (e+f x)}{d}\right )^m \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}\right ) \text {Subst}\left (\int \frac {\left (\frac {x}{d}\right )^{-m} \left (1+\frac {b x^2}{a}\right )^p}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {F_1\left (\frac {1-m}{2};1,-p;\frac {3-m}{2};-\tan ^2(e+f x),-\frac {b \tan ^2(e+f x)}{a}\right ) (d \cot (e+f x))^m \tan (e+f x) \left (a+b \tan ^2(e+f x)\right )^p \left (1+\frac {b \tan ^2(e+f x)}{a}\right )^{-p}}{f (1-m)}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(107)=214\).
time = 1.73, size = 265, normalized size = 2.48 \begin {gather*} -\frac {a (-3+m) F_1\left (\frac {1-m}{2};-p,1;\frac {3-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cos ^2(e+f x) \cot (e+f x) (d \cot (e+f x))^m \left (a+b \tan ^2(e+f x)\right )^p}{f (-1+m) \left (-2 b p F_1\left (\frac {3-m}{2};1-p,1;\frac {5-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+2 a F_1\left (\frac {3-m}{2};-p,2;\frac {5-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right )+a (-3+m) F_1\left (\frac {1-m}{2};-p,1;\frac {3-m}{2};-\frac {b \tan ^2(e+f x)}{a},-\tan ^2(e+f x)\right ) \cot ^2(e+f x)\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \left (d \cot \left (f x +e \right )\right )^{m} \left (a +b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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